A botanical garden is installing a new irrigation system and has three main supply hoses m… — State PSC Exam

A botanical garden is installing a new irrigation system and has three main supply hoses measuring $\frac{44}{5}$ meters, $\frac{77}{4}$ meters, and $\frac{121}{10}$ meters in length. The landscaping team needs to cut all three hoses into smaller, equal-length segments to connect to individual planters. If no material can be wasted, what is the minimum total number of segments that can be produced from all three hoses combined?

Cevap: 73 segments

Cevap

73 segments

The minimum number of total segments is achieved when each segment is cut to its maximum possible equal length. This maximum length is the HCF of the three fractions (44/5, 77/4, 121/10), which evaluates to 11/20 meters. Dividing the original lengths by this HCF yields exactly 16, 35, and 22 pieces respectively, totaling 73 individual segments.

Adım Adım Çözüm

Determine the mathematical requirement for minimizing the number of segments.

Identify that the Highest Common Factor (HCF) of the hose lengths must be found.

To get the minimum number of pieces, each piece must be of the maximum possible equal length.

Apply the rule for finding the HCF of fractions.

Use the formula:

\text{HCF} = \frac{\text{HCF of numerators}}{\text{LCM of denominators}}

This formula allows the calculation of the greatest common divisor for non-integer fractional values.

Calculate the HCF of the numerators.

The numerators are

44

77

, and

121

. Their HCF is

11

The largest integer that perfectly divides

44

77

, and

121

11

(

44 = 11 \times 4

77 = 11 \times 7

121 = 11 \times 11

Calculate the LCM of the denominators.

The denominators are

5

4

, and

10

. Their LCM is

20

The smallest integer that is a multiple of

5

4

, and

10

20

Determine the maximum segment length.

The segment length is

\frac{11}{20}

meters.

Combining the results from the previous steps using the fraction HCF formula yields the length.

Calculate the number of segments produced from each hose.

First hose:

\frac{44}{5} \div \frac{11}{20} = 16

. Second hose:

\frac{77}{4} \div \frac{11}{20} = 35

. Third hose:

\frac{121}{10} \div \frac{11}{20} = 22

Dividing the total length of each hose by the length of one segment gives the segment count per hose.

Sum the segment counts.

16 + 35 + 22 = 73

segments.

The question asks for the minimum total number of segments produced from all three hoses combined.

Anahtar Kavram

Calculating the Highest Common Factor (HCF) of fractions and applying it to optimize division in real-world scenarios.

Tahmini Süre:2m 0s

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